Ideals, Varieties, and Algorithms

An Introduction to Computational Algebraic Geometry and Commutative Algebra

  • David Cox
  • John Little
  • Donal O’Shea

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xv
  2. David Cox, John Little, Donal O’Shea
    Pages 1-48
  3. David Cox, John Little, Donal O’Shea
    Pages 49-114
  4. David Cox, John Little, Donal O’Shea
    Pages 115-168
  5. David Cox, John Little, Donal O’Shea
    Pages 169-214
  6. David Cox, John Little, Donal O’Shea
    Pages 215-264
  7. David Cox, John Little, Donal O’Shea
    Pages 265-316
  8. David Cox, John Little, Donal O’Shea
    Pages 317-356
  9. David Cox, John Little, Donal O’Shea
    Pages 357-438
  10. David Cox, John Little, Donal O’Shea
    Pages 439-508
  11. Back Matter
    Pages 509-551

About this book


Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?

The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes:

A significantly updated section on Maple in Appendix C

Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR

A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3

From the 2nd Edition:

"I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." -The American Mathematical Monthly


Maple Mathematica addition algebra commutative property computer algebra system proof

Authors and affiliations

  • David Cox
    • 1
  • John Little
    • 2
  • Donal O’Shea
    • 3
  1. 1.Department of Mathematics and Computer ScienceAmherst CollegeAmherstUSA
  2. 2.Department of Mathematics and Computer ScienceCollege of the Holy CrossWorcesterUSA
  3. 3.Department of Mathematics and StatisticsMount Holyoke CollegeSouth HadleyUSA

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